The Removal Property for Linear Configurations in Compact Abelian Groups

The combinatorial removal lemma states that, if a (hyper)graph K has not many copies of the fixed (hyper)graph H , then K can be made free of copies of H by removing a small set of (hyper)edges from K. In this thesis we show results for homomorphisms in finite abelian groups, and for integer linear systems over compact abelian groups that are analogous to the combinatorial removal lemma. The results state that, given some subsets Xi of the group, if there are not many solutions to the system Ax = 0, where the variables xi take values inXi, then there exist small subsets Xi' inside Xi such that there is no solution to the system Ax = 0 with xi in Xi \ Xi'. These results are shown by constructing an appropriate (hyper)graph that allows us to retrieve information on the elements to be removed from Xi using the set of (hyper)edges. These arithmetic removal lemmas extend the first removal lemma for groups proved by Green in 2005. They also present a comprehensive approach to results involving finding non-trivial linear configurations in dense sets. Examples of these types of results are SzemerĂŠdi's Theorem, which ensures the existence of arbitrarily long arithmetic progressions in sets of the integers with positive upper density, or its multidimensional version. The latter was first shown by Furstenberg and Katznelson in 1978 using ergodic theory and proves the existence of k-simplices in subsets containing a positive proportion of the integer lattice or rank k.Ph.D